# Questions tagged [lie-groupoids]

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### Differentiation of Lie $\infty$-groupoids

I've been trying to understand how to differentiate Lie $\infty$-groupoids to get a Lie $\infty$-algebroid. First of all, I will state the definitions that I'm assuming.
A Lie $\infty$-groupoid is a ...

**5**

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139 views

### On Holonomy in (regular) Riemannian Foliations

Right now, I am trying to understanding the role of holonomy fields on Riemannian foliations, which lead me to the following (probably topological) groupoid:
Let $\mathcal{F}\subset M$ be a ...

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339 views

### KK-witnesses of Gysin maps between differentiable stacks

In 1982 Alain Connes gave the construction of a KK-element $f! \in KK(C(X), C(Y))$ that "witnesses" the fiber integration/Gysin/Umkehr/wrong-way map on topological $K$-theory along a K-orientable map ...

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114 views

### Is this groupoid a model for the derived fixed-point locus of the free loop space?

In this paper, John Baez and Urs Schreiber define (see Definition 2.16) a Lie groupoid (there called a '2-space') associated to any manifold $M$. In fact it is a bundle of Lie groups over $M$ thought ...

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262 views

### Morita Equivalence of Lie groupoids

I am trying to understand what exactly is the Morita equivalence of Lie groupoids.
I am reading Ieke Moerdijk’s notes Orbifolds as Groupoids.
A homomorphism $\phi:\mathcal{H}\rightarrow \mathcal{G}...

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105 views

### Properties of the induced map between inertia stacks

Let $\mathcal X$ and $\mathcal Y$ be (separated) Deligne-Mumford stacks. A morphism of stacks $f:\mathcal X \to \mathcal Y$ induces a morphism between inertia stacks $\tilde f:I\mathcal X \to I\...

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100 views

### Action of a Lie groupoid on a Lie Algebroid?

Let $\pi:E\longrightarrow M$ be a vector bundle. Then we can associate a Lie groupoid $\mathsf{Gl}(E)\rightrightarrows M$ where $$\mathsf{Gl}(E):=\{E_x\stackrel{lin. isom.}{\longrightarrow} E_y: x, y\...

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230 views

### Which makes Lie groupoids so nice?

This is a continuation of my previous question.
A) Morphisms in (1') are basically internal anafunctors, their compositions heavily use (and only) pullback/limit.
B) Bibundles in (2) are basically ...

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85 views

### Idea behind definition of classifying space over an orbifold

Today I was explaining to some one the notion of $\mathcal{G}$ spaces, covering spaces over orbifolds from Orbifolds as Groupoids: an Introduction.
Definition : Let $X$ be a locally compact ...

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141 views

### adjoint representation of 2-Lie groups

Baez and Crans in their paper on Lie 2-algebras refer to adjoint representations of Lie 2-groups but don't say much, as far as I can tell, except to say that such a representation acts on a 2-Lie ...

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### Lie groupoid with non trivial automorphisms does not come from genuine space

This question came from comments of this question.
Suppose $\mathcal{G}$ is a Lie groupoid, we consider the category $B\mathcal{G}$ whose objects are principal $\mathcal{G}$ bundles and morphisms are ...